It depends.  If your new resident state allows for the pro-ration of the income that was earned prior to the move, you can adjust this in TurboTax during the state interview.  Be sure that you indicate that you were only a part-year resident of the state and, if your state has one, file the part-year resident form.

If your new state allows only for credit for taxes paid to another state, you may have less option to adjust.  However, even many of those states allow you to exclude income that was earned prior to the move. 

To help TurboTax make sure that in the My Info section of the program, you have indicated that you lived in more than one state for the year.  To check this click on My Info in the menu on the left, then click Edit beside your name.  You are looking for a screen that looks like this: 

The general rule is that the income is taxable when and where received.  If you received the money in 2019, it's taxable by the state you lived in in 2019, even though some of the money was back pay for a previous year.